Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500262 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24058 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #61664 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
843 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 843 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,11)(4,15)(5,18)(6,20)(9,23)(10,25)(12,28)(14,24)(16,21)(17,22)(26,29);(1,2,7,4)(3,12,11,26)(5,15,18,8)(6,17,20,10)(9,24,16,14)(13,28,27,29)(19,22,31,25)(21,30,23,32);(1,3,5,13,7,11,18,27)(2,9,8,21,4,16,15,23)(6,14,19,30,20,24,31,32)(10,12,22,29,17,26,25,28);(1,4,7,2)(3,9,11,16)(5,8,18,15)(6,17,20,10)(12,24,26,14)(13,23,27,21)(19,22,31,25)(28,32,29,30)Polycyclic presentation:
2^9|b^2=i,c^2=e,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,f^b=fh,g^b=gi,d^c=dh,f^c=fi,h^c=hi,e^d=ei,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407462);